The Schottky problem is a classical question asking for a characterization of Jacobians  among abelian varieties. In positive characteristic,  one may asking whether there are values of discreet invariants known to occurs for abelian varieties which do not occur for Jacobians. In this talk, I will focus on one such invariant, the Newton polygon, and discuss an approach to this question which relies on the study of the geometry of Shimura varieties intersecting the Torelli locus.

This talk is based on joint work with Li and Singh.